How to Solve the PDE 2Uxxy + 3Uxyy - Uxy = 0?

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Discussion Overview

The discussion revolves around solving the partial differential equation (PDE) given by 2Uxxy + 3Uxyy - Uxy = 0, where U is a function of x and y. Participants explore different methods and approaches to find a solution, including substitutions and changes of coordinates.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant made a substitution W=Uxy and applied a change of coordinates to reduce the PDE to a simpler form, seeking further assistance on the next steps.
  • Another participant proposed a general solution to the PDE, expressing it in terms of arbitrary functions F1, F2, and F3.
  • A request for clarification was made regarding the derivation of the proposed general solution.
  • A different perspective was offered, suggesting that the general solution can also be expressed through double integration of the transformed function, asserting that this method is equivalent to the previously mentioned solution.

Areas of Agreement / Disagreement

Participants have differing views on the methods to solve the PDE, with no consensus on a single approach. Multiple competing views remain regarding the general solution and the techniques used to arrive at it.

Contextual Notes

There are unresolved assumptions regarding the functions F1, F2, and F3, as well as the specifics of the integration process mentioned by one participant. The discussion does not clarify the conditions under which the proposed solutions hold.

mglaros
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2Uxxy+3Uxyy-Uxy=0 where U=U(x,y)

I made the substitution W=Uxy and then used a change of coordinates (n= 2x+3y, and r=3x-2y) which reduced the problem to solving Uxy=f(3x-2y)exp((2x+3y)/3) because W=f(r)exp(n/3). Now I have no idea where to go from there. Any help would be much appreciated.

Thanks
 
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The general solution to your PDE is as follows

U(x,y) = F1(x)+F2(y)+exp(x/2)F3(3x-2y),

where F1, F2, F3 are arbitrary functions.
 
kosovtsov,

Would you mind explaining to me how you arrived at that solution? It would be greatly appreciated.

Thanks
 
There is a general principle. If a problem can be solved, it as a rule, can be solved by countless number of methods.

For example, from your

Uxy=f(3x-2y)exp((2x+3y)/3)

it is follows immediately by double integration that the general solution can be in form

U(x,y)=F1(x)+F2(y)+\int \int (f(3x-2y)exp((2x+3y)/3)) dx dy .

My solution only looks simpler, but is equivalent the one above.
 

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